Smallest number k such that kn + 1 is prime

Conjecture: for every n > 1 there exists a number k < n such that nk + 1 is a prime.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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Cite

@misc{cairn-oeis-34693,
  title        = {Smallest number k such that kn + 1 is prime},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-34693}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

exists_k. Conjecture: for every $n > 1$ there exists a number $k < n$ such that $nk + 1$ is a prime.

exists_k_stronger. A stronger conjecture: for every n there exists a number $k < 1 + n^{0.75}$ such that $nk + 1$ is a prime.

a_isBigO. Conjecture: $a (n) = O (lo g (n) lo g (lo g (n)))$ .

a_unbounded. Counter-conjecture to a_isBigO: $a (n) / (lo g n lo g lo g n)$ is unbounded.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«34693» (4 statements).

theorem exists_k {n : ℕ} (hn : 1 < n) : ∃ k < n, (n * k + 1).Prime

theorem exists_k_stronger {n : ℕ} (hn : 0 < n) : ∃ k : ℕ,
    k < 1 + (Real.nthRoot 4 n) ^ 3 ∧ (n * k + 1).Prime

theorem a_isBigO : (fun n ↦ (a n : ℝ)) =O[atTop] (fun n ↦ Real.log n * Real.log (Real.log n))

theorem a_unbounded : ¬BddAbove (Set.range fun n ↦ a n / (Real.log n * Real.log (Real.log n)))

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

A34693

Source and licence

Imported from Formal Conjectures (OEIS), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.